Generalized taxicab number

In mathematics, the generalized taxicab number Taxicab(k, j, n) is the smallest number which can be expressed as the sum of j kth positive powers in n different ways. For k = 3 and j = 2, they coincide with Taxicab numbers.

It has been shown by Euler that

mathrm{Taxicab}(4, 2, 2) = 635318657 = 59^4 + 158^4 = 133^4 + 134^4

However, Taxicab(5, 2, n) is not known for any n ≥ 2; No positive integer is known at all which can be written as the sum of two fifth powers in more than one way.

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